Stochastic independence in non-commutative probability theory

1979 ◽  
Vol 86 (1) ◽  
pp. 103-114
Author(s):  
Wulf Driessler ◽  
Ivan F. Wilde
Keyword(s):  
Probability Theory ◽  
Probability Space ◽  
Random Variables ◽  
Independent Random Variables ◽  
Characteristic Functions ◽  
Usual Sense ◽  
Stochastic Independence ◽  
Selfadjoint Operators

AbstractFor a family {Xα} of random variables over a probability space , stochastic independence can be formulated in terms of factorization properties of characteristic functions. This idea is reformulated for a family {Aα} of selfadjoint operators over a probability gage space and is shown to be inappropriate as a non-commutative generalization. Indeed, such factorization properties imply that the {Aα} mutually commute and are versions of independent random variables in the usual sense.

scholarly journals Some results on the span of families of Banach valued independent, random variables

1991 ◽  
Vol 14 (2) ◽  
pp. 381-384
Author(s):  
Rohan Hemasinha
Keyword(s):  
Banach Space ◽  
Probability Space ◽  
Linear Span ◽  
Random Variables ◽  
Isomorphic Copy ◽  
Independent Random Variables ◽  
Closed Linear Span

LetEbe a Banach space, and let(Ω,ℱ,P)be a probability space. IfL1(Ω)contains an isomorphic copy ofL1[0,1]then inLEP(Ω)(1≤P<∞), the closed linear span of every sequence of independent,Evalued mean zero random variables has infinite codimension. IfEis reflexive orB-convex and1<P<∞then the closed(in LEP(Ω))linear span of any family of independent,Evalued, mean zero random variables is super-reflexive.

Large correlated families of positive random variables

1988 ◽  
Vol 103 (1) ◽  
pp. 147-162 ◽  
Author(s):  
D. H. Fremlin
Keyword(s):  
Probability Space ◽  
Random Variables ◽  
Characteristic Functions ◽  
Measurable Functions ◽  
Uniformly Bounded

S. Argyros and N. Kalamidas([l], repeated in [2], Theorem 6·15) proved the following. If κ is a cardinal of uncountable cofinality, and 〈Eξ〉ξ<κ is a family of measurable sets in a probability space (X, μ) such that infξ<κμEξ = δ, and if n ≥ 1, , then there is a set Γ ⊆ κ such that #(Γ) = κ and μ(∩ξ∈IEξ) ≥ γ whenever I ⊆ ξ has n members. In Proposition 7 below I refine this result by (i) taking any γ < δn (which is best possible) and (ii) extending the result to infinite cardinals of countable cofinality, thereby removing what turns out to be an irrelevant restriction. The proof makes it natural to perform a further extension to general uniformly bounded families of non-negative measurable functions in place of characteristic functions.

scholarly journals On C.R. RAO’s theorem for locally compact abelian groups

2019 ◽  
Vol 484 (3) ◽  
pp. 273-276
Author(s):  
G. M. Feldman
Keyword(s):  
Abelian Group ◽  
Random Vector ◽  
Compact Abelian Group ◽  
Random Variables ◽  
Independent Random Variables ◽  
Characteristic Functions ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups

Let x1, x2, x3 be independent random variables with values in a locally compact Abelian group X with nonvanish- ing characteristic functions, and aj, bj be continuous endomorphisms of X satisfying some restrictions. Let L1 = a1x1 + a2x2 + a3x3, L2 = b1x1 + b2x2 + b3x3. It was proved that the distribution of the random vector (L1; L2) determines the distributions of the random variables xj up a shift. This result is a group analogue of the well-known C.R. Rao theorem. We also prove an analogue of another C.R. Rao’s theorem for independent random variables with values in an a-adic solenoid.

Negative probability

1945 ◽  
Vol 41 (1) ◽  
pp. 71-73 ◽  
Author(s):  
M. S. Bartlett
Keyword(s):  
Probability Theory ◽  
Quantum Theory ◽  
Physical Interpretation ◽  
Random Variables ◽  
Characteristic Functions ◽  
Negative Probability ◽  
Number Of Particles

SummaryIt has been shown that orthodox probability theory may consistently be extended to include probability numbers outside the conventional range, and in particular negative probabilities. Random variables are correspondingly generalized to include extraordiary random variables; these have been defined in general, however, only through their characteristic functions.This generalized theory implies redundancy, and its use is a matter of convenience. Eddington(3) has employed it in this sense to introduce a correction to the fluctuation in number of particles within a given volume.Negative probabilities must always be combined with positive ones to give an ordinary probability before a physical interpretation is admissible. This suggests that where negative probabilities have appeared spontaneously in quantum theory it is due to the mathematical segregation of systems or states which physically only exist in combination.

A characterization of the gamma distribution from a random difference equation

1988 ◽  
Vol 25 (1) ◽  
pp. 142-149 ◽  
Author(s):  
Eric S. Tollar
Keyword(s):  
Difference Equation ◽  
Gamma Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Characteristic Functions ◽  
Random Difference Equation ◽  
Random Difference

A characterization of the gamma distribution is considered which arises from a random difference equation. A proof without characteristic functions is given that if V and Y are independent random variables, then the independence of V · Y and (1 – V) · Y results in a characterization of the gamma distribution (after excluding the trivial cases).

A characterization of the gamma distribution from a random difference equation

1988 ◽  
Vol 25 (01) ◽  
pp. 142-149 ◽  
Author(s):  
Eric S. Tollar
Keyword(s):  
Difference Equation ◽  
Gamma Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Characteristic Functions ◽  
Random Difference Equation ◽  
Random Difference

A characterization of the gamma distribution is considered which arises from a random difference equation. A proof without characteristic functions is given that if V and Y are independent random variables, then the independence of V · Y and (1 – V) · Y results in a characterization of the gamma distribution (after excluding the trivial cases).

Order statistics of partial sums of mutually independent random variables

1975 ◽  
Vol 12 (02) ◽  
pp. 390-395 ◽  
Author(s):  
Felix Pollaczek
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Random Variables ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Characteristic Functions ◽  
Analytic Proof ◽  
Mutually Independent ◽  
Independent Identically Distributed

Herein is exposed a simplified analytic proof of formulas for the characteristic functions of ordered partial sums of mutually independent identically distributed random variables. This problem which we had raised and solved in 1952 by another method, has since been treated by several authors (see Wendel [6]), and recently by de Smit [4], who made use of a kind of Wiener-Hopf decomposition†. On the contrary our present as well as our previous proof essentially uses the explicit solution of a certain singular integral equation in a complex domain.

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